copyright 2011 pearson education, inc. slide 9.5-1


Chapter 9: Trigonometric Identities and Equations (I)

9.1 Trigonometric Identities

9.2 Sum and Difference Identities

9.3 Further Identities

9.4 The Inverse Circular Functions

9.5 Trigonometric Equations and Inequalities (I)

9.6 Trigonometric Equations and Inequalities (II)


9.5 Trigonometric Equations and Inequalities (I)

• Solving a Trigonometric Equation by Linear Methods

Example Solve 2 sin x – 1 = 0 over the interval [0, 2).

Analytic Solution Since this equation involves the first power of sin x, it is linear in sin x.

21

sin

1sin201sin2

x

xx

., isset solution theTherefore.20for sinsatisfy , and ,for valuesTwo

65

6

21

65

6

xxx


9.5 Solving a Trigonometric Equation by Linear Methods

Graphing Calculator SolutionGraph y = 2 sin x – 1 over the interval [0, 2].

The x-intercepts have the same decimal approximations as . and 6

56


9.5 Solving Trigonometric Inequalities

Example Solve for x over the interval [0, 2).(a) 2 sin x –1 > 0 and (b) 2 sin x –1 < 0.

Solution(a) Identify the values for which the graph of

y = 2 sin x –1 is above the x-axis. From the previous graph, the solution set is

(b) Identify the values for which the graph of y = 2 sin x –1 is below the x-axis. From the previous graph, the solution set is

., 65

6

.2,,0 65

6


9.5 Solving a Trigonometric Equation by Factoring

Example Solve sin x tan x = sin x over the interval [0°, 360°).

Solution

0)1(tansin0sintansinsintansin

xxxxx

xxx

0sin x1tan01tanor

xx

225or45180or0 xxxx

.225,180,45,0 isset solution The

Caution Avoid dividing both sides by sin x. The two solutions that make sin x = 0 would not appear.


9.5 Equations Solvable by Factoring

Example Solve tan2 x + tan x –2 = 0 over the interval [0, 2).Solution This equation is quadratic in term tan x.

The solutions for tan x = 1 in [0, 2) are x = Use a calculator to find the solution to tan-1(–2) –1.107148718. To get the values in the interval [0, 2), we add and 2 to tan-1(–2) to get

x = tan-1(–2) + 2.03443936 and x = tan-1(–2) + 2 5.176036589.

0)2)(tan1(tan02tantan2

xxxx

1tan01tan

xx

2tan02tan

x

xoror

. 45

4 or


9.5 Solving a Trigonometric Equation Using the Quadratic Formula

Example Solve cot x(cot x + 3) = 1 over the interval [0, 2).

Solution Rewrite the expression in standard quadratic

form to get cot2 x + 3 cot x – 1 = 0, with a = 1, b = 3, c = –1, and cot x as the variable.

Since we cannot take the inverse cotangent with the calculator, we use the fact that

2133

2493cot x

3027756377.cotor302775638.3cot xx

.cot tan1

xx


9.5 Solving a Trigonometric Equation Using the Quadratic Formula

The first of these, –.29400113018, is not in the desired interval. Since the period of cotangent is , we add and then 2 to –.29400113018 to get 2.847591352 and 5.989184005.

The second value, 1.276795025, is in the interval, so we add to it to get another solution.

The solution set is {1.28, 2.85, 4.42, 5.99}.

3027756377.cotor302775638.3cot xx

2940013018.x or 276795025.1x302775638.3tanor3027756377.tan xx


9.5 Solving a Trigonometric Equation by Squaring and Trigonometric Substitution

Example Solve over the interval [0, 2).

Solution Square both sides and use the identity 1 + tan2 x = sec2 x.

xx sec3tan

33

31

tan

2tan32tan13tan32tan

sec3tan32tansec3tan

22

22

x

xxxx

xxxxx

. issolution only t theVerify tha . and are solutions Possible

611

611

65


9.4 The Inverse Sine Function

Solving a Trigonometric Equation Analytically1. Decide whether the equation is linear or quadratic, so

you can determine the solution method.2. If only one trigonometric function is present, solve the

equation for that function.3. If more than one trigonometric function is present,

rearrange the equation so that one side equals 0. Then try to factor and set each factor equal to 0 to solve.

4. If the equation is quadratic in form, but not factorable, use the quadratic formula. Check that solutions are in the desired interval.

5. Try using identities to change the form of the equation. It may be helpful to square each side of the equation first. If this is done, check for extraneous values.


9.4 The Inverse Sine Function

Solving a Trigonometric Equation Graphically1. For an equation of the form f(x) = g(x), use the

intersection-of-graphs method.2. For an equation of the form f(x) = 0, use the

x-intercept method.

copyright 2011 pearson education, inc. slide 9.5-1

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